Does anyone know how to integrate the sqrt(1 + e^x) ? Maybe I just don't see it, but doing it by parts or trig substitution isn't working for me. Any tips? Thanks in advance.
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Does anyone know how to integrate the sqrt(1 + e^x) ? Maybe I just don't see it, but doing it by parts or trig substitution isn't working for me. Any tips? Thanks in advance.
Let I represent the integral to be solved. And let In{} represent the integral sign.
Now, use this substitution:
1 + e-x = p2
whereby,
-e-xdx = 2pdp
and
dx = 2pdp / -e-x = -2pdp / (p2 - 1)
Therefore:
I = -2*In{p2dp / (p2 - 1)}
By simple algebra this can be decomposed into:
I = -In{p / (p + 1)} - In{p / (p - 1)}
and these two are easy enough (for example, substitute further p + 1 = v in the first term and p - 1 = w in the second). The rest should be straightforward.
By the way, the correct solution is:
I = Ln{[Sqrt(1 + e-x) + 1] / [Sqrt(1 + e-x) - 1]} - 2*Sqrt(1 + e-x)
where Ln means natural logarithm (base e)
I hope you don't get confused by the notation.
thanks! i never thought of that substitution.